Basic Concepts of the Theory of Sets
In discussing any branch of mathematics, be it analysis, algebra, or geometry, it is helpful to u the notation and terminology of t theory. This subject, which was developed by Boole and Cantor in the latter part of the 19th century, has had a profound influence on the development of mathematics in the 20th century. It has unified many emingly disconnected ideas and has helped to reduce many mathematical concepts to their logical foundations in an elegant and systematic way. A thorough treatment of theory of ts would require a lengthy discussion which we regard as outside the scope of this book. Fortunately, the basic noticns are few in number, and it is possible to develop a working knowledge of the methods and ideas of t theory through an informal discussion . Actually, we shall discuss not so much a new th金骏眉属于什么茶
eory as an agreement about the preci terminology that we wish to apply to more or less familiar ideas.
In mathematics, the word “t” is ud to reprent a collection of objects viewed as a single entity
The collections called to mind by such nouns as “flock”, “tribe”, ‘crowd”, “team’, are all examples of ts, The individual objects in the collection are called elements or members of the t, and they are said to belong to or to be contained in the t. The t in turn ,is said to contain or be compod of its elements.
We shall be interested primarily in ts of mathematical objects: ts of numbers, ts of curves, ts of geometric figures, and so on. In many applications it is convenient to deal with ts in which nothing special is assumed about the nature of the individual objects in the collection. The are called abstract ts. Abstract t theory has been developed to deal with such collections of arbitrary objects, and from this generality the theory derives its power.
NOTATIONS. Sets usually are denoted by capital letters: A,B,C,….X,Y,Z ; elements are designated by lower-ca letters: a, b, c,….x, y, z. We u the special notation
X∈S
To mean that “x is an element of S “or” x belongs to S”. If x does not belong to S, we write x∈S. When convenient ,we shall designate ts by displaying the elements in braces; for example ，the t of positive even integers less than 10 is denoted by the symbol{2,4,6,8}whereas the t of all positive even integers is displayed as {2,4,6,…},the dots taking the place of “and so on”.
The first basic concept that relates one t to another is equality of ts:
DEFINITION OF SET EQUALITY Two ts A and B are said to be equal(or identical)if they consist of exactly the same elements, in which ca we write A=B. If one of the ts contains an element not in the other ,we say the ts are unequal and we write A≠B.
SUBSETS. From a given t S we may form new ts, called subts of S. For example, the t consisting of tho positive integers less than 10 which are divisible by 4(the t{湿气拍打方法
4,8})is a subt of the t of all even integers less than 10.In general, we have the following definition.
DEFINITION OF A SUBSET.A t A is said to be a subt of a t B, and we write
A B
Whenever every element of A also belongs to B. We also say that A is contained in B or B contains A. The relatio上课的英语单词
n is referred to as t inclusion.
The statement A B does not rule out the possibility t毽球
hat B A. In fact, we may have both A B and B A, but this happens only if A and B have the same elements. In other words, A=B if and only if A B and B A .
This theorem is an immediate conquence of the foregoing definitions of equality and inclusion. If A B but A≠B, then we say that A is a proper subt of B: we indicate this by writing A B.
In all our applications of t theory, we have a fixed t S given in advance, and we are concerned only with subts of this given t. The underlying t S may vary from one application to another; it will be referred to as the universal t of each particular discour.
The notation
{X∣X∈S. and X satisfies P}
will designate the t of all elements X in S which satisfy the property P. When the universal t to which w中山大学简介
e are referring id understood, we omit the reference to S and we simply write{X∣X satisfies P}.This is read “the t of all x such that x satisfies p.” Sets designated in this way are said to be described by a defining property For example, the t of all positive real numbers could be designated as {X∣X>0};the universal t S in this ca is understood to be the t of all real numbers. Of cour, the letter x is a dummy and may be replaced by any other convenient symbol. Thus we may write
